| import OAI.MeasureTheory.Falconer.Recursion.ProfileGrid |
| import OAI.MeasureTheory.Falconer.Estimates.AllDimensionalOccupiedClasses |
| import OAI.MeasureTheory.Falconer.Estimates.RegularizationScales |
|
|
| namespace OAI |
|
|
| /-! |
| # Mass-profile comparison at every depth in arbitrary dimension |
|
|
| The interpolation loss is (d+1)*lambda. The block depths satisfy the |
| same numerical bracketing as in the planar case. |
| -/ |
|
|
| namespace Falconer.Regularization |
| variable {d : ℕ} |
| open MeasureTheory |
| open scoped ENNReal |
|
|
| noncomputable def sourceProfile (Q : DyadicCube d) (μ : Measure (Space d)) |
| (N B : ℕ) (p : CellPath d) : ℝ → ℝ := |
| regularizedProfile (regularizationLeaves Q μ N) (regularizationLabel Q μ N B) N B (N/B+1) p |
|
|
| noncomputable def sourceClass (Q : DyadicCube d) (μ : Measure (Space d)) |
| (N B : ℕ) (p : CellPath d) : Measure (Space d) := |
| normalizedClass Q μ (regularizationLeaves Q μ N) (regularizationLabel Q μ N B) |
| (regularizationLabel Q μ N B p) |
|
|
| theorem sourceClass_grid_comparison (Q : DyadicCube d) (hQ : 0 < Q.side) |
| (μ : Measure (Space d)) [IsProbabilityMeasure μ] {N B : ℕ} (hN : 0 < N) (hB : 0 < B) |
| {p : CellPath d} (hp : p ∈ regularizationLeaves Q μ N) |
| {i : ℕ} (hi : i ≤ N/B+1) {P : CellPath d} |
| (hP : P ∈ (classLeaves (regularizationLeaves Q μ N) (regularizationLabel Q μ N B) |
| (regularizationLabel Q μ N B p)).image (List.take (blockDepth N B i))) : |
| (((2:ℝ)^N)^(-sourceProfile Q μ N B p (blockDepth N B i)))/(2:ℝ)^(N/B+2) ≤ |
| (sourceClass Q μ N B p (Q.descendant P).carrier).toReal ∧ |
| (sourceClass Q μ N B p (Q.descendant P).carrier).toReal ≤ |
| (2:ℝ)^(N/B+2)*(((2:ℝ)^N)^(-sourceProfile Q μ N B p (blockDepth N B i))) := by |
| classical |
| rw [sourceProfile,regularizedProfile_reference_grid hN (show regularizationLeaves Q μ N ⊆ cubePaths d N from Finset.filter_subset _ _) |
| (regularizationLabel Q μ N B) hB p hi] |
| exact regularization_grid_mass_bounds Q hQ μ N hB hp hi hP |
|
|
| theorem profile_bracket_mass_bound {R C lam m₀ mₙ m₁ x₀ xₙ x₁ : ℝ} |
| (hR : 1 ≤ R) (hC : 0 < C) (hCL : C ≤ R^lam) |
| (hml : m₁-mₙ ≤ (d:ℝ)*lam) (hmu : mₙ-m₀ ≤ (d:ℝ)*lam) |
| (hlo : R^(-m₁)/C ≤ x₁) (hhi : x₀ ≤ C*R^(-m₀)) |
| (hxl : x₁ ≤ xₙ) (hxu : xₙ ≤ x₀) : |
| R^(-((d:ℝ)+1)*lam)*R^(-mₙ) ≤ xₙ ∧ xₙ ≤ R^(((d:ℝ)+1)*lam)*R^(-mₙ) := by |
| have hRp : 0 < R := lt_of_lt_of_le zero_lt_one hR |
| constructor |
| · apply le_trans ?_ (hlo.trans hxl) |
| apply (le_div_iff₀ hC).mpr |
| calc |
| R^(-((d:ℝ)+1)*lam)*R^(-mₙ)*C ≤ R^(-((d:ℝ)+1)*lam)*R^(-mₙ)*R^lam := |
| mul_le_mul_of_nonneg_left hCL (by positivity) |
| _ = R^(-(d:ℝ)*lam-mₙ) := by rw [← Real.rpow_add hRp,← Real.rpow_add hRp]; congr 1; ring |
| _ ≤ R^(-m₁) := Real.rpow_le_rpow_of_exponent_le hR (by linarith) |
| · apply le_trans (hxu.trans hhi) |
| calc |
| C*R^(-m₀) ≤ R^lam*R^(-m₀) := mul_le_mul_of_nonneg_right hCL (by positivity) |
| _ = R^(lam-m₀) := by rw [← Real.rpow_add hRp]; rfl |
| _ ≤ R^(((d:ℝ)+1)*lam-mₙ) := Real.rpow_le_rpow_of_exponent_le hR (by linarith) |
| _ = R^(((d:ℝ)+1)*lam)*R^(-mₙ) := Real.rpow_add hRp _ _ |
|
|
| theorem sourceProfile_short_increment (Q : DyadicCube d) (μ : Measure (Space d)) |
| {N B : ℕ} (hN : 0 < N) (hB : 0 < B) (p : CellPath d) {lam u v : ℝ} |
| (hwidth : (B:ℝ) ≤ lam*(N:ℝ)) (hu : 0 ≤ u) (huv : u ≤ v) (hv : v ≤ N) |
| (hshort : v-u ≤ B) : |
| sourceProfile Q μ N B p v-sourceProfile Q μ N B p u ≤ (d:ℝ)*lam := by |
| classical |
| have hh := regularizedProfile_increment hN (show regularizationLeaves Q μ N ⊆ cubePaths d N from Finset.filter_subset _ _) |
| (regularizationLabel Q μ N B) hB p hu huv hv |
| apply hh.2.trans |
| apply (div_le_iff₀ (by exact_mod_cast hN : 0 < (N:ℝ))).mpr |
| have hmul₁ := mul_le_mul_of_nonneg_left hwidth (Nat.cast_nonneg d) |
| have hmul₂ := mul_le_mul_of_nonneg_left hshort (Nat.cast_nonneg d) |
| nlinarith |
|
|
| theorem sourceClass_all_depth_comparison (Q : DyadicCube d) (hQ : 0 < Q.side) |
| (μ : Measure (Space d)) [IsProbabilityMeasure μ] {N B : ℕ} (hN : 0 < N) (hB : 0 < B) |
| {lam : ℝ} (hwidth : (B:ℝ) ≤ lam*(N:ℝ)) |
| (hlarge : (2:ℝ)^(N/B+2) ≤ ((2:ℝ)^N)^lam) |
| {p : CellPath d} (hp : p ∈ regularizationLeaves Q μ N) |
| {n : ℕ} (hn : n ≤ N) {P : CellPath d} |
| (hP : P ∈ (classLeaves (regularizationLeaves Q μ N) (regularizationLabel Q μ N B) |
| (regularizationLabel Q μ N B p)).image (List.take n)) : |
| ((2:ℝ)^N)^(-((d:ℝ)+1)*lam)*((2:ℝ)^N)^(-sourceProfile Q μ N B p n) ≤ |
| (sourceClass Q μ N B p (Q.descendant P).carrier).toReal ∧ |
| (sourceClass Q μ N B p (Q.descendant P).carrier).toReal ≤ |
| ((2:ℝ)^N)^(((d:ℝ)+1)*lam)*((2:ℝ)^N)^(-sourceProfile Q μ N B p n) := by |
| classical |
| let i := (N-n)/B |
| have hb := PlanarFalconer.blockDepth_brackets hn hB |
| change i < N/B+1 ∧ blockDepth N B (i+1) ≤ n ∧ n ≤ blockDepth N B i ∧ |
| blockDepth N B i-n ≤ B ∧ n-blockDepth N B (i+1) ≤ B at hb |
| obtain ⟨q,hq,rfl⟩ := Finset.mem_image.mp hP |
| let ρ := sourceClass Q μ N B p |
| have : IsProbabilityMeasure ρ := regularization_class_probability Q hQ μ N B hp |
| have hgp := sourceClass_grid_comparison Q hQ μ hN hB hp (Nat.le_of_lt hb.1) |
| (Finset.mem_image.mpr ⟨q,hq,rfl⟩) |
| have hgm := sourceClass_grid_comparison Q hQ μ hN hB hp (by omega : i+1 ≤ N/B+1) |
| (Finset.mem_image.mpr ⟨q,hq,rfl⟩) |
| have hmono₁ : ρ (Q.descendant (q.take (blockDepth N B i))).carrier ≤ |
| ρ (Q.descendant (q.take n)).carrier := by |
| apply measure_mono |
| have hh := descendant_subset_take Q hQ (q.take (blockDepth N B i)) n |
| simpa only [List.take_take,Nat.min_eq_left hb.2.2.1] using hh |
| have hmono₀ : ρ (Q.descendant (q.take n)).carrier ≤ |
| ρ (Q.descendant (q.take (blockDepth N B (i+1)))).carrier := by |
| apply measure_mono |
| have hh := descendant_subset_take Q hQ (q.take n) (blockDepth N B (i+1)) |
| simpa only [List.take_take,Nat.min_eq_left hb.2.1] using hh |
| have hml := sourceProfile_short_increment Q μ hN hB p hwidth |
| (Nat.cast_nonneg n) (Nat.cast_le.mpr hb.2.2.1) |
| (Nat.cast_le.mpr (Nat.sub_le N (i*B))) (by |
| simpa only [Nat.cast_sub hb.2.2.1] using |
| (show (((blockDepth N B i-n):ℕ):ℝ) ≤ (B:ℝ) by exact_mod_cast hb.2.2.2.1)) |
| have hmu := sourceProfile_short_increment Q μ hN hB p hwidth |
| (Nat.cast_nonneg (blockDepth N B (i+1))) (Nat.cast_le.mpr hb.2.1) (Nat.cast_le.mpr hn) (by |
| simpa only [Nat.cast_sub hb.2.1] using |
| (show (((n-blockDepth N B (i+1)):ℕ):ℝ) ≤ (B:ℝ) by exact_mod_cast hb.2.2.2.2)) |
| exact profile_bracket_mass_bound (one_le_pow₀ (by norm_num : (1:ℝ) ≤ 2)) |
| (by positivity) hlarge hml hmu hgp.1 hgm.2 |
| (ENNReal.toReal_mono (measure_ne_top ρ _) hmono₁) |
| (ENNReal.toReal_mono (measure_ne_top ρ _) hmono₀) |
|
|
| end Falconer.Regularization |
|
|
| end OAI |
|
|
